I am reading a paper which title is the same as the title of this question. Demonstration of Lemma 3.13 in the paper says that
$ P\left ( \frac{\left \|Ax^{(j)} \right \|}{\left \| x^{(j)} \right \|} < \mu\left \| A \right \| \right )=P\left ( \frac{\left \|Ax \right \|}{\left \| x \right \|} < \mu\left \| A \right \| \right ) $
just because the random vectors $x^{(1)},x^{(2)}...x^{(j)}$ are supposed to be distributed the same as the vector $x$. If I were to say that the equality holds for $x^{(1)}$ and $x^{(2)}$ I would be lying since they are taken randomly from a gaussian distribution and they are not equal, therefore the function of this random variable $\left \| Ax \right \|$ will be different while the constant $\mu\left \| A \right \|$ is the same. But when it comes to a general vector $x$ then somehow all the probabilities are equal. Quite frankly I don't see why having equal distributions on the vectors implies the equalitys shown. Anyone willing to point me to a textbook explaining these kinds of calculations involving probability with matrices will have my sincerest thanks.
Thank you
You seem to be confusing the random variable with its observed value.
If $X$ and $Y$ are random variables (for example, $X$ is the result of the next roll of a six-sided die, and $Y$ is the result of the roll after that), then a statement like $P(X\ge 4) = P(Y\ge 4)$ is true whenever $X$ and $Y$ have the same distribution.
But statement is false if you plug in the observed values for $X$ and $Y$. If the next roll of the die turns out to be $2$ and the one after that turns out to be $5$, then we have observed the value $2$ for $X$ and the value $4$ for $Y$. It is certainly false that $P(2\ge4) = P(5\ge4)$, but the assertion $P(X\ge4)=P(Y\ge4)$ does hold for the random variables $X$ and $Y$, since they have the same distribution.
The equality you've described is another instance of this principle. It's saying that since the random vector $x^{(j)}$ has the same distribution as the random vector $x$, any event involving $x^{(j)}$ has the same probability as the corresponding event involving $x$. The fact that we are working with matrices is immaterial.